Atlas › Modular forms › Weight 2 › Level 903 › 903.2.a

Newspace 903.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(903))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
903.2.a.a	1	$\mathbb{Q}$	$q + q^{3} - 2q^{4} - q^{7} + O(q^{8})$	− + −	$-1$	903a1 rank 1
903.2.a.b	1	$\mathbb{Q}$	$q + q^{3} - 2q^{4} + q^{7} + O(q^{8})$	− − −	$+1$	903b1 rank 0
903.2.a.c	2	$\mathbb{Q}(\sqrt{5})$	$2q - 3q^{2} - 2q^{3} + 3q^{4} - 3q^{5} + 3q^{6} - 2q^{7} + O(q^{8})$	+ + +	$-1$
903.2.a.d	2	$\mathbb{Q}(\sqrt{5})$	$2q - q^{2} + 2q^{3} - q^{4} - 3q^{5} - q^{6} + 2q^{7} + O(q^{8})$	− − +	$-1$
903.2.a.e	2	$\mathbb{Q}(\sqrt{13})$	$2q - q^{2} + 2q^{3} + 3q^{4} + q^{5} - q^{6} - 2q^{7} + O(q^{8})$	− + −	$-1$
903.2.a.f	2	$\mathbb{Q}(\sqrt{3})$	$2q - 2q^{3} + 2q^{4} + 2q^{5} - 2q^{7} + O(q^{8})$	+ + +	$-1$
903.2.a.g	5	degree 5	$5q - 3q^{2} - 5q^{3} + 7q^{4} - 5q^{5} + 3q^{6} + 5q^{7} + O(q^{8})$	+ − −	$-1$
903.2.a.h	5	degree 5	$5q + 2q^{2} - 5q^{3} + 4q^{4} + 7q^{5} - 2q^{6} + 5q^{7} + O(q^{8})$	+ − +	$+1$
903.2.a.i	7	degree 7	$7q + 4q^{2} + 7q^{3} + 10q^{4} + 5q^{5} + 4q^{6} - 7q^{7} + O(q^{8})$	− + +	$+1$
903.2.a.j	8	degree 8	$8q + q^{2} + 8q^{3} + 13q^{4} + 3q^{5} + q^{6} + 8q^{7} + O(q^{8})$	− − −	$+1$
903.2.a.k	8	degree 8	$8q + 2q^{2} - 8q^{3} + 12q^{4} + 3q^{5} - 2q^{6} - 8q^{7} + O(q^{8})$	+ + −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 2T_{5} + 3T_{11} + 4T_{13} + 5T_{17} + 6T_{19} + 7T_{23} + 8T_{29}$ generates each orbit's coefficient field; on the newspace its characteristic polynomial factors over $\mathbb{Q}$ as one irreducible factor per orbit:

Orbit	Characteristic polynomial of $T$
903.2.a.a	$x + 109$
903.2.a.b	$x - 119$
903.2.a.c	$x^{2} + 34x - 431$
903.2.a.d	$x^{2} + 62x - 1919$
903.2.a.e	$x^{2} + 114x - 5539$
903.2.a.f	$x^{2} + 100x + 1417$
903.2.a.g	$x^{5} + 151x^{4} - 10948x^{3} - 1951532x^{2} - 39113013x + 64488989$
903.2.a.h	$x^{5} - 219x^{4} + 14314x^{3} - 71836x^{2} - 21047323x + 499881871$
903.2.a.i	$x^{7} - 327x^{6} + 37767x^{5} - 1988663x^{4} + 53030527x^{3} - 708208697x^{2} + 3946375017x - 2914385993$
903.2.a.j	$x^{8} + 54x^{7} - 9116x^{6} - 379284x^{5} + 17183182x^{4} + 836010386x^{3} + 3395881772x^{2} - 148332912388x - 1005470545439$
903.2.a.k	$x^{8} - 130x^{7} - 5254x^{6} + 720896x^{5} + 3230920x^{4} - 1010417514x^{3} + 11095895862x^{2} + 99667603756x - 1135001887033$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(903))$	43
Old forms	70
Cusp forms $S_{2}(\Gamma_0(903))$	113
Eisenstein series $E_{2}(\Gamma_0(903))$	7
Modular forms $M_{2}(\Gamma_0(903))$	120

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(903)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(21))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(43))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(129))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(301))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.697 s (one thread) and proven: the dimensions are certified, and the characteristic polynomial of $T$ over $\mathbb{Z}$ is reconstructed by CRT past a rigorous coefficient bound.

The engine's checks

dim M^+ at levels [1, 3, 7, 21, 43, 129, 301, 903]: [0, 1, 1, 4, 4, 16, 30, 120] (certified at ell = 2147483647)
T = [(2, 1), (5, 2), (11, 3), (13, 4), (17, 5), (19, 6), (23, 7), (29, 8)]; h of degree 43 = 1 x 43 (phi(ord eps) x dim S_k^new); 12 primes, 0 rejected

Open in the notebook · JSON · stored data

Recompute it yourself

In Sage mode at sagebrush.space (or npx sagebrush --sage):

newform_orbits(903, 2)

With the Rust engine (source), in engine/:

cargo run --release -p sagebrush-web --example atlas -- 903:2 1000

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